In this paper, two important structural properties, i.e., strong observability and strong detectability, are introduced for linear hybrid systems with periodic jumps. These properties are characterized in terms of geometric and algebraic conditions over the system matrices. The concepts and characterizations of the weakly unobservable subspace and the hybrid invariant zeros are also introduced, respectively. An algorithm to compute the weakly unobservable subspace is provided. In addition, it is shown that there exists a close relationship between the hybrid invariant zeros and the properties ofstrong observability and strong detectability. Some examples illustrate the proposed properties.